A Special Function

Number Theory Level 3

$a$ § $b$ is a special function involving $a$ and $b$ . If the following statements are true,

3§5 = 7

7§2 = 36

12§6 = 21

What is $a$ if $9$ § $a = 31$ ?

The answer is 3.

1 solution

Stewart Feasby
Oct 7, 2014

This was fun to construct actually.

a§b is simply defined as: $\overline{ab} \div b$ Where $\overline{xy} = 10x + y$

We see this as in No.1, $35\div 5 = 7$

No.2, $72\div 2 = 36$

No.3, $126\div 6 = 21$

Therefore, 9§a = 31, can be rewritten as: $\overline{9a} \div a = 31$ $\frac{90 + a}{a} = 31$ $90+a=31a$ $90=30a$ $a=\boxed {3}$ Although this problem is not entirely clear, if you can spot this pattern it makes it a heck load easier!

FYI - 3 terms is often not enough to establish a pattern.

Calvin Lin Staff - 6 years, 7 months ago

Oh sorry, how many is recommended for a pattern to be established.

Stewart Feasby - 6 years, 7 months ago

It generally depends. Pattern recognition is tricky, because there are always infinitely many ways to continue it, and then substantiate it with a certain polynomial.

In this case, it is not too bad since we see the 3 5 and 7, and are likely to think 35 is a multiple of 7.

If the function was more complicated / not easily observed, then more examples are needed. For example, if the question was
$5 \heartsuit 3 = 7$ , $2 \heartsuit 7 = 36$ , $6 \heartsuit 12 = 21$ , then it would be much harder to guess that $a \heartsuit b = \frac{ 10b + a } { a }$ , even though it is "essentially" the same question as you stated.

Calvin Lin Staff - 6 years, 7 months ago

@Calvin Lin – Ah ok. Thanks. I'll keep that in mind next time I make a similar problem!

Stewart Feasby - 6 years, 7 months ago

A Special Function

The answer is 3.

1 solution

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